Your latest edit asks a different question than has been asked previously. I have made some more edits to bring your question more in line with your recent focus and reopened your question.
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called2voyage♦Dec 12 '13 at 14:52

2 Answers
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the Angular Momentum Conservation Law states that, for any moving body, its angular momentum does not change unless you exercise an external force different from the central force.

For an orbiting body like a planet, this means that Sun's gravity, being the central force, does not modify Angular Momentum, but any other external force will do.

Examples of external forces are collisions or the forces made by Jupiter on another planet, or by Neptune on Pluto.

After the Solar System was formed, these external forces are quite small, and thus does not change greatly the Angular Momentum of any major body. But you can see how passing near a body can alter a comet's orbit.

Moreover, the external forces made by bodies that are in the same plane as an orbiting body does modify the value of its Angular Momentum, but not the direction. This causes that the orbiting body changes its orbit but can not not change planes.

So if you add small forces from objects in the same plane, you end up with no changes to planes.

Angular momentum conservation

To put it in more mathematical terms, you can play with the energy and the angular momentum of a bunch of particles orbinting a central mass $M$, given by

$$E = \sum_i m_i \left(\frac{1}{2}v_i^2 - \frac{GM}{r_i}\right),$$

for the energy and

$${\bf I} = \sum_i m_i {\bf r}_i \times {\bf v_i},$$

for the angular momentum. Now, let's try to extremize the energy for a given angular momentum, keeping in mind that the system has to conserve angular momentum, and that collisions between the particles can reduce the energy. One good way to do it is to use Lagrange multiplier